HW1 Submission

Damanpreet Singh

Andrew ID: damanprs

q1 - Affine Rectification

For the q1, the algorithm used is as follows:

• Select points on the image, to create set of parallel lines
• Project these points into projective space, by appending 1 at the end of each
• Generate lines by using the formula $ L = \overrightarrow{p1} \times \overrightarrow{p2} $
• Generate point at infinty using the parallel lines, as $p_{1inf} = \overrightarrow{L1}\times\overrightarrow{L2}$, similarly for $p_{2inf}$
• Generate line at infinty as, $l_{inf} = \overrightarrow{p_{1inf}} \times \overrightarrow{p_{2inf}}$
• As for affine rectification, the homography matrix is, $H = \begin{bmatrix}1&0&0\\0&1&0\\l_1&l_2&1\end{bmatrix} $, we solve the equation of $H^{-T}l_{inf} = \begin{bmatrix}0\\0\\1\end{bmatrix}$
• This leads to a solution of $l_1 = a_2 , l_2 = a_1$, where $l_{inf} = \begin{bmatrix}a_1\\a_2\\1\end{bmatrix}$

Results

tiles5

Input Image	Annotated parallel lines	Affine-Rectified Image

Line	Before	After
Green	0.933045	0.999662
Blue	0.999998	0.999997

tiles4

Input Image	Annotated parallel lines	Affine-Rectified Image

Line	Before	After
Green	0.9802158	0.999982
Blue	0.999894	0.9999371

tiles6

Input Image	Annotated parallel lines	Affine-Rectified Image

Line	Before	After
Green	0.999890	0.999768
Blue	0.9995473	0.9999913

calib1 (Calibration board)

Input Image	Annotated parallel lines	Affine-Rectified Image

Line	Before	After
Green	0.97703	1.0
Blue	0.999798	0.999833

res_building (Residential Building)

Input Image	Annotated parallel lines	Affine-Rectified Image

Line	Before	After
Green	0.999358	0.9990105
Blue	0.998932	0.9997795

q2 - Metric Rectification

For q2, the algorithm used is as follows:

• Two sets of perpendicular lines are constructed using the same method as q1
• The general shape of a line conic for an affine transformed image is $C_{\infty}^* = \begin{bmatrix} a&b&0\\b&c&0\\0&0& 0\end{bmatrix}$
• Using a set of perpendicular lines, we know $l C_{\infty}^* m = 0$, we can construct a system of equations as $Ac=0$ , to extract $a,b,c$
• These values are extracted using SVD
• We need to compute an $H$, such that
• This is accomplished using SVD , as $ C_{\infty}^* = UDU^T $
• Here, final H is calculated as, $H = \begin{bmatrix} \sqrt{\sigma_1^{-1}} & 0&0 \\ 0&\sqrt{\sigma_2^{-1}}&0\\0&0&1 \end{bmatrix}U^T$
• So, the whole algorithm invloves two SVDs

Results

tiles5

Input Image	Annotated parallel lines on affine image	Metric-Rectified Image

Line	Before	After
Green	-0.195470	-0.0054723
Blue	0.0847146	0.0108689

tiles4

Input Image	Annotated parallel lines on affine image	Metric-Rectified Image

Line	Before	After
Green	0.3796108	0.1708844
Blue	0.35701868	0.203533

tiles6

Input Image	Annotated parallel lines on affine image	Metric-Rectified Image

Line	Before	After
Green	0.09290548	0.14334456
Blue	0.106657065	-0.15170103

q3 - Planar Homography

For q3, the algorithm used is:

• Take 4 corner points of the first image
• Take 4 corner points in the second image, on the positon you want to warp the first image

• As we know, $Hp_1 \times p_2 =0$, we develop a $2n\times9$ matrix $A$, as

  

• This gives rise to a system of equation $Ah = 0$, wher $h$ is the fllattened homegraphy matrix $H$

• The value is calculated using SVD
• The first image is warped using the $\textit{MyWarp}$ function
• The warped image is then copied into a bigger image, to fit the size of the second image
• The area where the warped image is to be overlayed in the second image, is cleared
• This clearing is done by detting co-ordinates of non-zero pixels in the warped image and putting those to $0$ in second image
• Finally, the warped image is overlaid onto the second image, using $\textit{cv2.add()}$

Results

Normal Image	Perspective Image	Annotated corners in perspective image	Warped and overlaid image

Late days